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Is it possible for an AC generator to actually generate electricity let's say being in a ship orbiting Mars or on the surface of the moon? Does Faraday's principle require the magnetig field of the earth?

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Faraday's law is one of Maxwell's equations, which are fundamental. It says that a changing magnetic field produces an electric field, no matter where you are. It works just as well in empty space as it does on the earth; the earth's magnetic field is not needed. A typical generator has its own permanent magnets inside, and they are sufficient.

https://electronics.howstuffworks.com/how-to-build-an-electic-generator.htm

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